metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: SD32⋊3D5, D10.6D8, D8.5D10, C16.11D10, Q16.2D10, C40.19C23, C80.11C22, Dic5.25D8, D40.3C22, Dic20.4C22, C4.7(D4×D5), (D5×C16)⋊5C2, C5⋊3(C4○D16), C5⋊D16⋊4C2, C5⋊Q32⋊2C2, C16⋊D5⋊6C2, C2.22(D5×D8), D8⋊3D5⋊5C2, (C5×SD32)⋊4C2, (C4×D5).60D4, C20.13(C2×D4), C10.38(C2×D8), C5⋊2C8.26D4, Q8.D10⋊4C2, (C5×D8).5C22, C8.25(C22×D5), C5⋊2C16.6C22, (C8×D5).41C22, (C5×Q16).3C22, SmallGroup(320,543)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for SD32⋊3D5
G = < a,b,c,d | a16=b2=c5=d2=1, bab=a7, ac=ca, ad=da, bc=cb, dbd=a8b, dcd=c-1 >
Subgroups: 422 in 84 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2 [×3], C4, C4 [×3], C22 [×3], C5, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], D5 [×2], C10, C10, C16, C16, C2×C8, D8, D8, SD16 [×2], Q16, Q16, C4○D4 [×2], Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C16, D16, SD32, SD32, Q32, C4○D8 [×2], C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20 [×2], C2×Dic5, C5⋊D4, C5×D4, C5×Q8, C4○D16, C5⋊2C16, C80, C8×D5, D40, Dic20, D4.D5, Q8⋊D5, C5×D8, C5×Q16, D4⋊2D5, Q8⋊2D5, D5×C16, C16⋊D5, C5⋊D16, C5⋊Q32, C5×SD32, D8⋊3D5, Q8.D10, SD32⋊3D5
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], C2×D8, C22×D5, C4○D16, D4×D5, D5×D8, SD32⋊3D5
►On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)
(1 67)(2 74)(3 65)(4 72)(5 79)(6 70)(7 77)(8 68)(9 75)(10 66)(11 73)(12 80)(13 71)(14 78)(15 69)(16 76)(17 55)(18 62)(19 53)(20 60)(21 51)(22 58)(23 49)(24 56)(25 63)(26 54)(27 61)(28 52)(29 59)(30 50)(31 57)(32 64)(33 115)(34 122)(35 113)(36 120)(37 127)(38 118)(39 125)(40 116)(41 123)(42 114)(43 121)(44 128)(45 119)(46 126)(47 117)(48 124)(81 109)(82 100)(83 107)(84 98)(85 105)(86 112)(87 103)(88 110)(89 101)(90 108)(91 99)(92 106)(93 97)(94 104)(95 111)(96 102)(129 145)(130 152)(131 159)(132 150)(133 157)(134 148)(135 155)(136 146)(137 153)(138 160)(139 151)(140 158)(141 149)(142 156)(143 147)(144 154)
(1 37 17 151 83)(2 38 18 152 84)(3 39 19 153 85)(4 40 20 154 86)(5 41 21 155 87)(6 42 22 156 88)(7 43 23 157 89)(8 44 24 158 90)(9 45 25 159 91)(10 46 26 160 92)(11 47 27 145 93)(12 48 28 146 94)(13 33 29 147 95)(14 34 30 148 96)(15 35 31 149 81)(16 36 32 150 82)(49 133 101 77 121)(50 134 102 78 122)(51 135 103 79 123)(52 136 104 80 124)(53 137 105 65 125)(54 138 106 66 126)(55 139 107 67 127)(56 140 108 68 128)(57 141 109 69 113)(58 142 110 70 114)(59 143 111 71 115)(60 144 112 72 116)(61 129 97 73 117)(62 130 98 74 118)(63 131 99 75 119)(64 132 100 76 120)
(1 106)(2 107)(3 108)(4 109)(5 110)(6 111)(7 112)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 62)(26 63)(27 64)(28 49)(29 50)(30 51)(31 52)(32 53)(33 134)(34 135)(35 136)(36 137)(37 138)(38 139)(39 140)(40 141)(41 142)(42 143)(43 144)(44 129)(45 130)(46 131)(47 132)(48 133)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 81)(113 154)(114 155)(115 156)(116 157)(117 158)(118 159)(119 160)(120 145)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 152)(128 153)
G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,67)(2,74)(3,65)(4,72)(5,79)(6,70)(7,77)(8,68)(9,75)(10,66)(11,73)(12,80)(13,71)(14,78)(15,69)(16,76)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,115)(34,122)(35,113)(36,120)(37,127)(38,118)(39,125)(40,116)(41,123)(42,114)(43,121)(44,128)(45,119)(46,126)(47,117)(48,124)(81,109)(82,100)(83,107)(84,98)(85,105)(86,112)(87,103)(88,110)(89,101)(90,108)(91,99)(92,106)(93,97)(94,104)(95,111)(96,102)(129,145)(130,152)(131,159)(132,150)(133,157)(134,148)(135,155)(136,146)(137,153)(138,160)(139,151)(140,158)(141,149)(142,156)(143,147)(144,154), (1,37,17,151,83)(2,38,18,152,84)(3,39,19,153,85)(4,40,20,154,86)(5,41,21,155,87)(6,42,22,156,88)(7,43,23,157,89)(8,44,24,158,90)(9,45,25,159,91)(10,46,26,160,92)(11,47,27,145,93)(12,48,28,146,94)(13,33,29,147,95)(14,34,30,148,96)(15,35,31,149,81)(16,36,32,150,82)(49,133,101,77,121)(50,134,102,78,122)(51,135,103,79,123)(52,136,104,80,124)(53,137,105,65,125)(54,138,106,66,126)(55,139,107,67,127)(56,140,108,68,128)(57,141,109,69,113)(58,142,110,70,114)(59,143,111,71,115)(60,144,112,72,116)(61,129,97,73,117)(62,130,98,74,118)(63,131,99,75,119)(64,132,100,76,120), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,129)(45,130)(46,131)(47,132)(48,133)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,81)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,152)(128,153)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160), (1,67)(2,74)(3,65)(4,72)(5,79)(6,70)(7,77)(8,68)(9,75)(10,66)(11,73)(12,80)(13,71)(14,78)(15,69)(16,76)(17,55)(18,62)(19,53)(20,60)(21,51)(22,58)(23,49)(24,56)(25,63)(26,54)(27,61)(28,52)(29,59)(30,50)(31,57)(32,64)(33,115)(34,122)(35,113)(36,120)(37,127)(38,118)(39,125)(40,116)(41,123)(42,114)(43,121)(44,128)(45,119)(46,126)(47,117)(48,124)(81,109)(82,100)(83,107)(84,98)(85,105)(86,112)(87,103)(88,110)(89,101)(90,108)(91,99)(92,106)(93,97)(94,104)(95,111)(96,102)(129,145)(130,152)(131,159)(132,150)(133,157)(134,148)(135,155)(136,146)(137,153)(138,160)(139,151)(140,158)(141,149)(142,156)(143,147)(144,154), (1,37,17,151,83)(2,38,18,152,84)(3,39,19,153,85)(4,40,20,154,86)(5,41,21,155,87)(6,42,22,156,88)(7,43,23,157,89)(8,44,24,158,90)(9,45,25,159,91)(10,46,26,160,92)(11,47,27,145,93)(12,48,28,146,94)(13,33,29,147,95)(14,34,30,148,96)(15,35,31,149,81)(16,36,32,150,82)(49,133,101,77,121)(50,134,102,78,122)(51,135,103,79,123)(52,136,104,80,124)(53,137,105,65,125)(54,138,106,66,126)(55,139,107,67,127)(56,140,108,68,128)(57,141,109,69,113)(58,142,110,70,114)(59,143,111,71,115)(60,144,112,72,116)(61,129,97,73,117)(62,130,98,74,118)(63,131,99,75,119)(64,132,100,76,120), (1,106)(2,107)(3,108)(4,109)(5,110)(6,111)(7,112)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,54)(18,55)(19,56)(20,57)(21,58)(22,59)(23,60)(24,61)(25,62)(26,63)(27,64)(28,49)(29,50)(30,51)(31,52)(32,53)(33,134)(34,135)(35,136)(36,137)(37,138)(38,139)(39,140)(40,141)(41,142)(42,143)(43,144)(44,129)(45,130)(46,131)(47,132)(48,133)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,81)(113,154)(114,155)(115,156)(116,157)(117,158)(118,159)(119,160)(120,145)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,152)(128,153) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)], [(1,67),(2,74),(3,65),(4,72),(5,79),(6,70),(7,77),(8,68),(9,75),(10,66),(11,73),(12,80),(13,71),(14,78),(15,69),(16,76),(17,55),(18,62),(19,53),(20,60),(21,51),(22,58),(23,49),(24,56),(25,63),(26,54),(27,61),(28,52),(29,59),(30,50),(31,57),(32,64),(33,115),(34,122),(35,113),(36,120),(37,127),(38,118),(39,125),(40,116),(41,123),(42,114),(43,121),(44,128),(45,119),(46,126),(47,117),(48,124),(81,109),(82,100),(83,107),(84,98),(85,105),(86,112),(87,103),(88,110),(89,101),(90,108),(91,99),(92,106),(93,97),(94,104),(95,111),(96,102),(129,145),(130,152),(131,159),(132,150),(133,157),(134,148),(135,155),(136,146),(137,153),(138,160),(139,151),(140,158),(141,149),(142,156),(143,147),(144,154)], [(1,37,17,151,83),(2,38,18,152,84),(3,39,19,153,85),(4,40,20,154,86),(5,41,21,155,87),(6,42,22,156,88),(7,43,23,157,89),(8,44,24,158,90),(9,45,25,159,91),(10,46,26,160,92),(11,47,27,145,93),(12,48,28,146,94),(13,33,29,147,95),(14,34,30,148,96),(15,35,31,149,81),(16,36,32,150,82),(49,133,101,77,121),(50,134,102,78,122),(51,135,103,79,123),(52,136,104,80,124),(53,137,105,65,125),(54,138,106,66,126),(55,139,107,67,127),(56,140,108,68,128),(57,141,109,69,113),(58,142,110,70,114),(59,143,111,71,115),(60,144,112,72,116),(61,129,97,73,117),(62,130,98,74,118),(63,131,99,75,119),(64,132,100,76,120)], [(1,106),(2,107),(3,108),(4,109),(5,110),(6,111),(7,112),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,54),(18,55),(19,56),(20,57),(21,58),(22,59),(23,60),(24,61),(25,62),(26,63),(27,64),(28,49),(29,50),(30,51),(31,52),(32,53),(33,134),(34,135),(35,136),(36,137),(37,138),(38,139),(39,140),(40,141),(41,142),(42,143),(43,144),(44,129),(45,130),(46,131),(47,132),(48,133),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,81),(113,154),(114,155),(115,156),(116,157),(117,158),(118,159),(119,160),(120,145),(121,146),(122,147),(123,148),(124,149),(125,150),(126,151),(127,152),(128,153)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 10 | 40 | 2 | 5 | 5 | 8 | 40 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D10 | C4○D16 | D4×D5 | D5×D8 | SD32⋊3D5 |
| kernel | SD32⋊3D5 | D5×C16 | C16⋊D5 | C5⋊D16 | C5⋊Q32 | C5×SD32 | D8⋊3D5 | Q8.D10 | C5⋊2C8 | C4×D5 | SD32 | Dic5 | D10 | C16 | D8 | Q16 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 |
Matrix representation of SD32⋊3D5 ►in GL4(𝔽241) generated by
| 138 | 41 | 0 | 0 |
| 200 | 138 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 214 | 156 | 0 | 0 |
| 156 | 27 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 51 | 1 |
| 0 | 0 | 240 | 0 |
| 0 | 177 | 0 | 0 |
| 64 | 0 | 0 | 0 |
| 0 | 0 | 240 | 190 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [138,200,0,0,41,138,0,0,0,0,1,0,0,0,0,1],[214,156,0,0,156,27,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,51,240,0,0,1,0],[0,64,0,0,177,0,0,0,0,0,240,0,0,0,190,1] >;
SD32⋊3D5 in GAP, Magma, Sage, TeX
{\rm SD}_{32}\rtimes_3D_5 % in TeX
G:=Group("SD32:3D5"); // GroupNames label
G:=SmallGroup(320,543);
// by ID
G=gap.SmallGroup(320,543);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,135,184,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^16=b^2=c^5=d^2=1,b*a*b=a^7,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^8*b,d*c*d=c^-1>;
// generators/relations